In the 19th century, German chemist and physicist
Julius von Mayer
Julius Robert von Mayer (25 November 1814 – 20 March 1878) was a German physician, chemist, and physicist and one of the founders of thermodynamics. He is best known for enunciating in 1841 one of the original statements of the conservation ...
derived a relation between
specific heat
In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
at constant pressure and the specific heat at constant volume for an
ideal gas
An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
. Mayer's relation states that
:
,
where is the
molar specific heat
Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K).
Heat capacity ...
at constant
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
, is the molar specific heat at constant
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
and is the
gas constant
The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
.
For more general homogeneous substances, not just ideal gases, the difference takes the form,
:
(see
relations between heat capacities
In thermodynamics, the heat capacity at constant volume, C_, and the heat capacity at constant pressure, C_, are extensive properties that have the magnitude of energy divided by temperature.
Relations
The laws of thermodynamics imply the follow ...
), where
is the
molar volume
In chemistry and related fields, the molar volume, symbol ''V''m, or \tilde V of a substance is the ratio of the volume occupied by a substance to the amount of substance, usually given at a given temperature and pressure. It is equal to the molar ...
,
is the temperature,
is the
thermal expansion coefficient
Thermal expansion is the tendency of matter to change its shape, area, volume, and density in response to a change in temperature, usually not including phase transitions.
Temperature is a monotonic function of the average molecular kinetic ...
and
is the isothermal
compressibility
In thermodynamics and fluid mechanics, the compressibility (also known as the coefficient of compressibility or, if the temperature is held constant, the isothermal compressibility) is a measure of the instantaneous relative volume change of a f ...
.
From this latter relation, several inferences can be made:
* Since the isothermal compressibility
is positive for nearly all phases, and the square of thermal expansion coefficient
is always either a positive quantity or zero, the specific heat at constant pressure is nearly always greater than or equal to specific heat at constant volume:
:
≥
.
:There are no known exceptions to this principle for gases or liquids, but certain solids are known to exhibit negative compressibilities
and presumably these would be (unusual) cases where
<
.
* For
incompressible
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An eq ...
substances, and are identical. Also for substances that are nearly incompressible, such as solids and liquids, the difference between the two specific heats is negligible.
* As the
absolute temperature
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.
Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wor ...
of the system approaches zero, since both heat capacities must generally approach zero in accordance with the
Third Law of Thermodynamics
The third law of thermodynamics states, regarding the properties of closed systems in thermodynamic equilibrium: This constant value cannot depend on any other parameters characterizing the closed system, such as pressure or applied magnetic fiel ...
, the difference between and also approaches zero. Exceptions to this rule might be found in systems exhibiting
residual entropy
Residual entropy is the difference in entropy between a non-equilibrium state and crystal state of a substance close to absolute zero. This term is used in condensed matter physics to describe the entropy at zero kelvin of a glass or plastic cry ...
due to disorder within the crystal.
References
{{Reflist
Thermodynamic equations